A307719 -id:A307719 - cp's OEIS frontend

A337602 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

0, 0, 0, 1, 3, 6, 10, 9, 18, 16, 24, 21, 43, 24, 51, 31, 54, 42, 94, 45, 102, 55, 99, 69, 163, 66, 150, 88, 168, 96, 265, 93, 228, 121, 246, 126, 337, 132, 315, 169, 342, 162, 487, 165, 420, 217, 411, 213, 619, 207, 558, 259, 540, 258, 784, 264, 654, 325, 660

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,3,3)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,5,1)  (1,3,4)
                    (2,1,2)  (1,4,1)  (2,2,3)  (1,4,3)
                    (2,2,1)  (2,1,3)  (2,3,2)  (1,5,2)
                    (3,1,1)  (2,2,2)  (3,1,3)  (1,6,1)
                             (2,3,1)  (3,2,2)  (2,1,5)
                             (3,1,2)  (3,3,1)  (2,3,3)
                             (3,2,1)  (5,1,1)  (2,5,1)
                             (4,1,1)           (3,1,4)
                                               (3,2,3)
                                               (3,3,2)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

The complement in A014311 of A337695 ranks these compositions.
A220377*6 is the strict case.
A337600 is the unordered version.
A337603 does not consider a singleton to be coprime unless it is (1).
A337664 counts these compositions of any length.
A000740 counts relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
A000217 counts 3-part compositions.
A001399/A069905/A211540 count 3-part partitions.
A023023 counts relatively prime 3-part partitions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime 3-part compositions.
Cf. A000212, A007359, A087087, A284825, A302696, A304709, A304712, A307719, A328673, A335235, A335238, A337483, A337562, A337601.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,100}]

A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17

Offset: 1

Vladeta Jovovic, Nov 28 2004

First differs from A018892 at a(30) = 15, A018892(30) = 14.
First differs from A343654 at a(210) = 51, A343654(210) = 52.
Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.
In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021

			From _Gus Wiseman_, May 01 2021: (Start)
The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)   (1,1,1)
           (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)   (1,1,2)
                    (1,1,4)  (1,1,3)  (1,1,4)  (1,1,3)   (1,1,3)
                             (1,1,6)  (1,1,8)  (1,1,4)   (1,1,4)
                             (1,2,3)           (1,1,6)   (1,1,6)
                                               (1,2,3)   (1,1,8)
                                               (1,3,4)   (1,2,3)
                                               (1,1,12)  (1,3,4)
                                                         (1,3,8)
                                                         (1,1,12)
                                                         (1,1,24)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A070919, A086222, A086165, A048691, A063647.
Positions of 2's through 5's are A000040, A001248, A030078, A068993.
The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The ordered version is A048785.
The strict case is A066620.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
The version for partitions is A307719 (no 1's: A337563).
The case of distinct parts coprime is A337600 (ordered: A337602).
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.

pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y,{2}]);
Table[Length[Select[Tuples[Divisors[n],3],LessEqual@@#&&pwcop[#]&]],{n,30}] (* Gus Wiseman, May 01 2021 *)

A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017

a(n) = (tau(n^3) + 3*tau(n) + 2)/6.

A337600 Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 9, 7, 10, 8, 11, 11, 18, 12, 19, 13, 19, 17, 30, 16, 28, 20, 31, 23, 47, 23, 42, 26, 45, 27, 60, 31, 57, 35, 61, 37, 85, 38, 75, 43, 74, 47, 108, 45, 98, 52, 96, 56, 136, 54, 115, 64, 117, 67, 175, 65, 139, 76, 144, 75, 195

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A337601 at a(9) = 5, A337601(9) = 4.

			The a(3) = 1 through a(14) = 10 partitions (A = 10, B = 11, C = 12):
  111  211  221  222  322  332  333  433  443  444  544  554
            311  321  331  431  441  532  533  543  553  743
                 411  511  521  522  541  551  552  661  752
                           611  531  721  722  651  733  761
                                711  811  731  732  751  833
                                          911  741  922  851
                                               831  B11  941
                                               921       A31
                                               A11       B21
                                                         C11

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A220377 is the strict case.
A304712 counts these partitions of any length.
A307719 is the strict case except for any number of 1's.
A337601 does not consider a singleton to be coprime unless it is (1).
A337602 is the ordered version.
A337664 counts compositions of this type and any length.
A000217 counts 3-part compositions.
A000837 counts relatively prime partitions.
A001399/A069905/A211540 count 3-part partitions.
A023023 counts relatively prime 3-part partitions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A304709 counts partitions whose distinct parts are pairwise coprime.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime length-3 compositions.
A337563 counts pairwise coprime length-3 partitions with no 1's.
Cf. A001840, A007359, A007360, A014612, A087087, A284825, A302569, A302696, A328673, A335235, A337603, A337695.

Table[Length[Select[IntegerPartitions[n,{3}],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,100}]

For n > 0, a(n) = A337601(n) + A079978(n).

A337664 Number of compositions of n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 30, 58, 111, 210, 396, 750, 1420, 2688, 5079, 9586, 18092, 34157, 64516, 121899, 230373, 435463, 823379, 1557421, 2946938, 5578111, 10561990, 20005129, 37902514, 71832373, 136173273, 258211603, 489738627, 929074448, 1762899110, 3345713034

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

			The a(0) = 1 through a(5) = 16 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220

A304712 is the unordered version.
A337562 is the strict case.
A337602 is the length-3 case.
A337665 does not consider a singleton to be coprime unless it is (1).
A337695 ranks the complement of these compositions.
A000740 counts relatively prime compositions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime length-3 compositions.
A337561 counts pairwise coprime strict compositions.
Cf. A007359, A007360, A087087, A302569, A304709, A307719, A335235, A335238, A335239, A337562, A337603, A337667.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,15}]

A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 4, 8, 8, 14, 10, 16, 12, 30, 38, 46, 46, 48, 52, 62, 152, 96, 156, 112, 190, 256, 338, 420, 394, 326, 402, 734, 622, 1150, 802, 946, 898, 1730, 1946, 2524, 2200, 2328, 2308, 3356, 5816, 4772, 5350, 4890, 6282, 6316, 12092, 8902

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. These compositions must be strict.

			The a(5) = 2 through a(12) = 14 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)  (3,7)    (2,9)  (5,7)
  (3,2)     (3,4)  (5,3)  (4,5)  (7,3)    (3,8)  (7,5)
            (4,3)         (5,4)  (2,3,5)  (4,7)  (2,3,7)
            (5,2)         (7,2)  (2,5,3)  (5,6)  (2,7,3)
                                 (3,2,5)  (6,5)  (3,2,7)
                                 (3,5,2)  (7,4)  (3,4,5)
                                 (5,2,3)  (8,3)  (3,5,4)
                                 (5,3,2)  (9,2)  (3,7,2)
                                                 (4,3,5)
                                                 (4,5,3)
                                                 (5,3,4)
                                                 (5,4,3)
                                                 (7,2,3)
                                                 (7,3,2)

A022340 intersected with A333227 is a ranking sequence (using standard compositions A066099) for these compositions.
A212804 does not require coprimality, with unordered version A002865.
A337450 is the relatively prime instead of pairwise coprime version, with strict case A337451 and unordered version A302698.
A337462 allows 1's, with strict case A337561 (or A101268 with singletons), unordered version A327516 with Heinz numbers A302696, and 3-part case A337461.
A337485 is the unordered version (or A007359 with singletons considered coprime), with Heinz numbers A337984.
A337563 is the case of unordered triples.
Cf. A078374, A178472, A302568, A302697, A305713, A307719, A332004, A337562.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}]

For n > 1, the version where singletons are considered coprime is a(n) + 1.

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

A338556 Products of three prime numbers of even index.

Original entry on oeis.org

27, 63, 117, 147, 171, 261, 273, 333, 343, 387, 399, 477, 507, 549, 609, 637, 639, 711, 741, 777, 801, 903, 909, 931, 963, 1017, 1083, 1113, 1131, 1179, 1183, 1251, 1281, 1359, 1421, 1443, 1467, 1491, 1557, 1629, 1653, 1659, 1677, 1729, 1737, 1791, 1813, 1869

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

All terms are odd.

Also Heinz numbers of integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
      27: {2,2,2}      637: {4,4,6}     1183: {4,6,6}
      63: {2,2,4}      639: {2,2,20}    1251: {2,2,34}
     117: {2,2,6}      711: {2,2,22}    1281: {2,4,18}
     147: {2,4,4}      741: {2,6,8}     1359: {2,2,36}
     171: {2,2,8}      777: {2,4,12}    1421: {4,4,10}
     261: {2,2,10}     801: {2,2,24}    1443: {2,6,12}
     273: {2,4,6}      903: {2,4,14}    1467: {2,2,38}
     333: {2,2,12}     909: {2,2,26}    1491: {2,4,20}
     343: {4,4,4}      931: {4,4,8}     1557: {2,2,40}
     387: {2,2,14}     963: {2,2,28}    1629: {2,2,42}
     399: {2,4,8}     1017: {2,2,30}    1653: {2,8,10}
     477: {2,2,16}    1083: {2,8,8}     1659: {2,4,22}
     507: {2,6,6}     1113: {2,4,16}    1677: {2,6,14}
     549: {2,2,18}    1131: {2,6,10}    1729: {4,6,8}
     609: {2,4,10}    1179: {2,2,32}    1737: {2,2,44}

A014612 allows all prime indices (not just even) (strict: A007304).
A066207 allows products of any length (strict: A258117).
A338471 is the version for odds instead of evens (strict: A307534).
A338557 is the strict case.
A014311 is a ranking of ordered triples (strict: A337453).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A005117 lists squarefree numbers, with even case A039956.
A008284 counts partitions by sum and length (strict: A008289).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A046316 lists products of exactly three odd primes (strict: A046389).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A307719 counts 3-part pairwise coprime partitions (strict: A220377).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337599, A337600.
Subsequence of A332820.

Select[Range[1000],PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338556(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))>>1)-(b>>1)+1 for a,k in filterfalse(lambda x:x[0]&1,enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2)) for b,m in filterfalse(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338471 Products of three prime numbers of odd index.

Original entry on oeis.org

8, 20, 44, 50, 68, 92, 110, 124, 125, 164, 170, 188, 230, 236, 242, 268, 275, 292, 310, 332, 374, 388, 410, 412, 425, 436, 470, 506, 508, 548, 575, 578, 590, 596, 605, 628, 668, 670, 682, 716, 730, 764, 775, 782, 788, 830, 844, 902, 908, 932, 935, 964, 970

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

Also Heinz numbers of integer partitions with 3 parts, all of which are odd. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
       8: {1,1,1}      268: {1,1,19}     575: {3,3,9}
      20: {1,1,3}      275: {3,3,5}      578: {1,7,7}
      44: {1,1,5}      292: {1,1,21}     590: {1,3,17}
      50: {1,3,3}      310: {1,3,11}     596: {1,1,35}
      68: {1,1,7}      332: {1,1,23}     605: {3,5,5}
      92: {1,1,9}      374: {1,5,7}      628: {1,1,37}
     110: {1,3,5}      388: {1,1,25}     668: {1,1,39}
     124: {1,1,11}     410: {1,3,13}     670: {1,3,19}
     125: {3,3,3}      412: {1,1,27}     682: {1,5,11}
     164: {1,1,13}     425: {3,3,7}      716: {1,1,41}
     170: {1,3,7}      436: {1,1,29}     730: {1,3,21}
     188: {1,1,15}     470: {1,3,15}     764: {1,1,43}
     230: {1,3,9}      506: {1,5,9}      775: {3,3,11}
     236: {1,1,17}     508: {1,1,31}     782: {1,7,9}
     242: {1,5,5}      548: {1,1,33}     788: {1,1,45}

Robert Israel, Table of n, a(n) for n = 1..10000

A066208 allows products of any length (strict: A258116).
A307534 is the squarefree case.
A338469 is the restriction to odds.
A338556 is the version for evens (strict: A338557).
A000009 counts partitions into odd parts (strict: A000700).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A008284 counts partitions by sum and length.
A014311 is a ranking of ordered triples (strict: A337453).
A014612 lists Heinz numbers of all triples (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A023023 counts 3-part relatively prime partitions (strict: A078374).
A046316 lists products of exactly three odd primes (strict: A046389).
A066207 lists numbers with all even prime indices (strict: A258117).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict triples.
A307719 counts 3-part pairwise coprime partitions (strict: A220377).
Cf. A000217, A001221, A001222, A005117, A037144, A056239, A112798.
Subsequence of A332820.

N:= 1000: # for terms <= N
R:= NULL:
for i from 1 by 2 do
  p:= ithprime(i);
  if p^3 >= N then break fi;
  for j from i by 2 do
    q:= ithprime(j);
    if p*q^2 >= N then break fi;
    for k from j by 2 do
      x:= p*q*ithprime(k);
      if x > N then break fi;
      R:= R,x;
od od od:
sort([R]); # Robert Israel, Jun 11 2025

Select[Range[100],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from sympy import primerange
from itertools import combinations_with_replacement as mc
def aupto(limit):
    pois = [p for i, p in enumerate(primerange(2, limit//4+1)) if i%2 == 0]
    return sorted(set(a*b*c for a, b, c in mc(pois, 3) if a*b*c <= limit))
print(aupto(971)) # Michael S. Branicky, Aug 20 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338471(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1)+1 for a,k in filter(lambda x:x[0]&1,enumerate(primerange(integer_nthroot(x,3)[0]+1),1)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338557 Products of three distinct prime numbers of even index.

Original entry on oeis.org

273, 399, 609, 741, 777, 903, 1113, 1131, 1281, 1443, 1491, 1653, 1659, 1677, 1729, 1869, 2067, 2109, 2121, 2247, 2373, 2379, 2451, 2639, 2751, 2769, 2919, 3021, 3081, 3171, 3219, 3367, 3423, 3471, 3477, 3633, 3741, 3801, 3857, 3913, 3939, 4047, 4053, 4173

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

All terms are odd.
Also sphenic numbers (A007304) with all even prime indices (A031215).
Also Heinz numbers of strict integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
     273: {2,4,6}     1869: {2,4,24}    3219: {2,10,12}
     399: {2,4,8}     2067: {2,6,16}    3367: {4,6,12}
     609: {2,4,10}    2109: {2,8,12}    3423: {2,4,38}
     741: {2,6,8}     2121: {2,4,26}    3471: {2,6,24}
     777: {2,4,12}    2247: {2,4,28}    3477: {2,8,18}
     903: {2,4,14}    2373: {2,4,30}    3633: {2,4,40}
    1113: {2,4,16}    2379: {2,6,18}    3741: {2,10,14}
    1131: {2,6,10}    2451: {2,8,14}    3801: {2,4,42}
    1281: {2,4,18}    2639: {4,6,10}    3857: {4,8,10}
    1443: {2,6,12}    2751: {2,4,32}    3913: {4,6,14}
    1491: {2,4,20}    2769: {2,6,20}    3939: {2,6,26}
    1653: {2,8,10}    2919: {2,4,34}    4047: {2,8,20}
    1659: {2,4,22}    3021: {2,8,16}    4053: {2,4,44}
    1677: {2,6,14}    3081: {2,6,22}    4173: {2,6,28}
    1729: {4,6,8}     3171: {2,4,36}    4179: {2,4,46}

For the following, NNS means "not necessarily strict".
A007304 allows all prime indices (not just even) (NNS: A014612).
A046389 allows all odd primes (NNS: A046316).
A258117 allows products of any length (NNS: A066207).
A307534 is the version for odds instead of evens (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338556 is the NNS version.
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers, with even case A039956.
A078374 counts 3-part relatively prime strict partitions (NNS: A023023).
A075819 lists even Heinz numbers of strict triples (NNS: A075818).
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
A258116 lists squarefree numbers with all odd prime indices (NNS: A066208).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337605.

Select[Range[1000],SquareFreeQ[#]&&PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (omega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, nextprime, integer_nthroot
def A338557(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))>>1)-(b>>1) for a,k in filterfalse(lambda x:x[0]&1,enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2)) for b,m in filterfalse(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338333 Number of relatively prime 3-part strict integer partitions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 7, 6, 10, 8, 14, 12, 18, 16, 24, 18, 30, 25, 34, 30, 44, 31, 52, 42, 56, 49, 69, 50, 80, 64, 83, 70, 102, 71, 114, 90, 112, 100, 140, 98, 153, 117, 153, 132, 184, 128, 195, 154, 196, 169, 234, 156, 252, 196, 241

Offset: 0

Gus Wiseman, Oct 30 2020

nonn

The Heinz numbers of these partitions are the intersection of A005117 (strict), A005408 (no 1's), A014612 (length 3), and A289509 (relatively prime).

			The a(9) = 1 through a(19) = 14 triples (A = 10, B = 11, C = 12, D = 13, E = 14):
  432   532   542   543   643   653   654   754   764   765   865
              632   732   652   743   753   763   854   873   874
                          742   752   762   853   863   954   964
                          832   932   843   943   872   972   973
                                      852   952   953   A53   982
                                      942   B32   962   B43   A54
                                      A32         A43   B52   A63
                                                  A52   D32   A72
                                                  B42         B53
                                                  C32         B62
                                                              C43
                                                              C52
                                                              D42
                                                              E32

A001399(n-9) does not require relative primality.
A005117 /\ A005408 /\ A014612 /\ A289509 gives the Heinz numbers.
A055684 is the 2-part version.
A284825 counts the case that is also pairwise non-coprime.
A337452 counts these partitions of any length.
A337563 is the pairwise coprime instead of relatively prime version.
A337605 is the pairwise non-coprime instead of relative prime version.
A338332 is the not necessarily strict version.
A338333*6 is the ordered version.
A000837 counts relatively prime partitions.
A008284 counts partitions by sum and length.
A078374 counts relatively prime strict partitions.
A101271 counts 3-part relatively prime strict partitions.
A220377 counts 3-part pairwise coprime strict partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A000741, A023022, A082024, A302698, A307719, A337450, A337599, A338468.

Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,30}]

cp's OEIS Frontend

A337602 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

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A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

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A337600 Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

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A337664 Number of compositions of n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

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A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

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A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

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A338556 Products of three prime numbers of even index.

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A338471 Products of three prime numbers of odd index.

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